arXiv subjects

Jakob Hofstad

Publications and source records attributed to Jakob Hofstad.

6 recordsLinked to original sources

arXiv2026

Concentration of the largest induced tree size of $G_{n,p}$ around the standard expectation threshold

Let $T(G)$ be the size of the largest induced tree of $G$, and let $G_{n,p}$ be the binomial random graph. Kamaldinov, Skorkin, and Zhukovskii proved that $T(G_{n,p})$ equals one of two consecutive values with high probability if $p$ is constant, and more recently, Oropeza extended this result to include all vanishing $p$ such that $p > n^{-\frac{e-2}{3e-2} + \epsilon}$, where $e$ is Euler's constant. We further extend this result to all vanishing $p$ such that $p \gg n^{-1/2} \ln^{3/2} n$, and furthermore, we show that, for $p$ such that $n^{-1} \ll p \ll n^{-1/2}, \ T(G_{n,p})$ cannot be concentrated at the standard expectation threshold.

math.CO↗

arXiv2024

A note on Two-Point Concentration of the Independence Number of $G_{n,m}$

We show that the independence number of $ G_{n,m}$ is concentrated on two values for $ n^{5/4+ ε} < m \le \binom{n}{2}$. This result establishes a distinction between $G_{n,m}$ and $G_{n,p}$ with $p = m/ \binom{n}{2}$ in the regime $ n^{5/4 + ε} < m< n^{4/3}$. In this regime the independence number of $ G_{n,m}$ is concentrated on two values while the independence number of $ G_{n,p}$ is not; indeed, for $p$ in this regime variations in $ α( G_{n,p})$ are determined by variations in the number of edges in $ G_{n,p}$.

math.CO↗

arXiv2024

Behavior of the Minimum Degree Throughout the $d$-process

The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$-process with $d$ fixed, with high probability, for each $j \in \{0,1,\dots,d-2\}$, the minimum degree jumps from $j$ to $j+1$ when the number of steps left is on the order of $\ln(n)^{d-j-1}$. This answers a question of Ruciński and Wormald. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln(n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.

math.CO↗

arXiv2024

Two-Point Concentration of the Independence Number of the Random Graph

We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ ε} < p \le 1$. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, concentrated on 2 values for $ p = o \left( (\log(n)/n)^{2/3} \right)$. The extent of concentration of the independence number of $ G_{n,p}$ for $ ω(1/n) <p \le n^{-2/3}$ remains an interesting open question.

math.CO↗

arXiv2024

A Critical Probability for Biclique Partition of $G_{n,p}$

The biclique partition number of a graph $G= (V,E)$, denoted $bp(G)$, is the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that $ bp(G) \leq n - α(G)$, where $α(G)$ is the maximum size of an independent set of $G$. Erdős conjectured in the 80's that for almost every graph $G$ equality holds; i.e., if $ G=G_{n,1/2}$ then $bp(G) = n - α(G)$ with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take $ G=G_{n,p}$, where $p$ is constant and less than a certain threshold value $p_0 \approx 0.312$. This verifies a conjecture of Chung and Peng for these values of $p$. We also show that if $p_0 < p <1/2$ then $bp(G_{n,p}) = n - (1 + Θ(1)) α(G_{n,p})$ with high probability.

math.CO↗

arXiv2019

Linear Factorization of Hypercyclic Functions for Differential Operators

On the Fréchet space of entire functions $H(\mathbb{C})$, we show that every nonscalar continuous linear operator $L:H(\mathbb{C})\to H(\mathbb{C})$ which commutes with differentiation has a hypercyclic vector $f(z)$ in the form of the infinite product of linear polynomials: \[ f(z) = \prod_{j=1}^\infty \, \left( 1-\frac{z}{a_j}\right), \] where each $a_j$ is a nonzero complex number.

math.FA↗